The Riemann Geometry and Its Generalization

VOL. 8, 1922

19

MATHEMATICS: EISENHART AND VEBLEN

In "Buckling" the woody cells are very weakly lignified. Other distinctions will doubtless reveal themselves on a more intensive study. Tetraploid plants from two distinct races were examined. Both showed an elongation of the trunk and an increase in the width of the first angle, the first pair of internodes and the stem diameter. Internally, the cells were found to be significantly larger than in the normal type and the woody tissue tended to be less strongly lignified. Only a preliminary survey of the field has as yet been made. The authors believe, however, that these Datura cultures provide exceptionally promising material for a study of the effect of specific factors and of specific chromosomes, particularly upon structural characters; and they hope through further investigation to be able to contribute materially to a factorial analysis of the chromosomes of this species. 1 Paper presented before the Botanical Society of America, December 28, 1921. Albert F. Blakeslee, "Types of Mutations and Their Possible Significance in Evolution," Amer. Naturalist, 55, 1921 (254-267); and "Variations in Datura Due to Changes in Chromosome Number," Ibid., Jan.-July, 1922. 2

THE RIEMANN GEOMETRY AND ITS GENERALIZATION By L. P. EISENHART AND 0. VEBLEN DEPARTMErNT OF MATHEMATICS, PRINCETON UMIVSRSITY Communicated. January 18, 1922

1. One of the simplest ways of generalizing Euclidean Geometry is to start by assuming (1) that the space to be considered is an n-dimensional manifold in the sense of Analysis Situs, and (2) that in this space there exists a system of curves called paths which, like the straight lines in a euclidean space, serve as a means of finding one's way about. These paths are defined as the solutions of a system of differential equations, dx? dxk d2xt (1.1) ds;+PJkdsdsO' in which the rji's are analytic functions of (x1, x2,....,x5) and the indices i, j, k run from 1 to n. The second term is a summation with regard to j and k in accordance with the usual convention in such formulas that any term represents a summation with respect to each letter which appears in it both as a subscript and as a superscript.

Since the second term is a quadratic form in

d-, there

is no loss of

generality in assuming, as we do, that

r,k =

(1.2)

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MA THEMA TICS: EISENHART AND VEBLEN PROC. N. A. S.

This definition of the paths is immediately suggested by the fact that the differential equations of the straight lines in a euclidean space which are

dS2x

(1.3) ds2X -0° in cartesian coordinates, take the form (1.1) in general coordinates, the r's now being such that there shall exist an analytic transformation of (xl, x2,.-..x.) converting (1.1) into (1.3). 2. This generalized geometry has been studied by H. Weyl in his book, "Raum, Zeit, Materie," Berlin, 1919, and in Vol. 1 of the "Mathematische Zeitschrift." It has also been considered by A. S. Eddington in Proc. Roy. Soc. London, 99A (1921). Both these authors define it in terms of a generalization of Levi-Civita's concept of infinitesimal parallelism rather than by the more natural idea of a system of paths. It reduces to the Riemann geometry if we assume that there exists a quadratic form dx' (2.1)

idx

(gij=gji)

with respect to which the paths are geodesics, i.e., if the curves for which the integral

CW ii

dxt dx' ds

(2.2)

ds ds differential equations (1.1). The conditions the satisfy stationary is that (2.2) be stationary are gaiI'7k

1l-?g

=

+

_gjk\

(2.3)

When solved for the derivatives of the g's these conditions become = °-(24 dg_ij~xk~~J1' gajrik -giXr.ik - gjT .(2.4) -

3. In general there exists no quadratic form (2.1) for which the paths are geodesics. For example, in a two-dimensional manifold the system of paths defined by d2x1 _

+ (X-X2)

(dx')2

) = 0,

(3.1) d2X2+ (x-x2) (-2

=

is one for which the equations (2.4) are inconsistent. Therefore there exists no quadratic form (2.1) of which the paths (3.1) are geodesics. The problem of determining under what conditions the geometry of

Voi,. 8, 1922

MA THEMA TICS: EISENHART A ND VEBLEN

21

paths is Riemannian is one of the inverse problems of the Calculus of Variations. But it does not seem hitherto to have been solved. Before considering it further we must mention a few general theorems, which have already been noted more or less explicitly by Weyl and Eddington. 4. If we put xi==.$(V,. . ... x), thus introducing a new set of coordinates, the equations (1.1) become

0, dxk

d+ irik ds ds ds2

O(4.1)

where

62XP

-x br q+r -._-

x

(4.2) Expressing the conditions of integrability of these equations regarded as differential equations for determining the x's as functions of the x's, when the r's and r's are known, we obtain bxJ x ax' BP= bxp 3t (4.3) where BPrs

=bxrq

_

aa

+

r

x

Equation (4.3) may also be written M tr' dxbx"ax'bxt lb

rqs

-as rqr. ,.

44

(4.5)

This states that the functions B are the components in the coordinate system x of a tensor, B, which is contravariant of the first order and covariant of the third order, with respect to the group of all transformations of the paths into themselves. This tensor is called the curvature tensor of the manifold. From (4.4) it follows that (4.6) BP + BP = 0 and (4.7) BPs + Bq +BPQ =0. The theory of covariant differentiation (cf. Ricci and Levi-Civita in Math. Ann., 54 (1901) can be generalized at once to the geometry of paths by replacing the Christoffel symbols {'k} by the functions r,k in all formulas. In particular it is easily proved by means of (4.1) and (4.2) that the operation of covariant differentiation converts any tensor into a tensor of higher order. The formulas for covariant differentiation of sums and products of tensors also generalize without change, and also the theorem that if aij is any tensor and aijkl is its second covariant derivative, (4.8) aijkl- aijlk aj Baiki + ai, Bjaki.

22

MATHEMATICS: EISENHARTAND VEBLEN

PROC. N. A. S.

5. Returning now to the question as to the conditions on the r's that they shall yield a Riemann geometry, we observe that the left hand member of (2.4) is the covariant derivative of gij. Thus (2.4) may be written (5.1) gijk = 0 . By (4.8) (5.2) gijkl - gijlk = g ikBsk + gia B7il which combined with (5.1) gives (5.3) g.j B' I + g,. B7ik, = 0. If these equations be differentiated covariantly, we get

gia Bjkim + gaj B' + gi.. B1k, + gajt Ba1 = Hence by (5.1) gia B7klm + gajBtim = 0. I.

0.

(5.4)

(5.5)

Proceeding in this manner we get a sequence of equations, namely gia B7kimn + gaj Bcklmn =0

(5.6) gia Biklmn*

+ gaj Bmn** p = °. If the geometry of paths is Riemannian (5.1) must be satisfied and hence also (5.3), (5.5) and (5.6). The equations (5.3), (5.5) and (5.6) are linear equations in the (n + 1)n/2 functions gij with coefficients which are functions of the r's alone. Hence the algebraic conditins for the consistency of (5.3), (5.5) and (5.6) regarded as linear equations in the g's are necessary conditions on the rs that the geometry of paths shall be Riemannian. 6. Now suppose that equations (5.3) and (5.5) are algebraically consistent in the g's, and that the rank of the matrix of the B's is such that the g's are determined by (5.3) to within a factor, which is at most a function of the x's. Let gjj stand for a particular solution of (5.3) and (5.5). If (5.3) be differential covariantly with respect to xm, we have in consequence of (5.5) (6.1) giam BY1ki + gajm Bckl = 0. Since these equations are of the same form as (5.3), it follows from the above hypothesis about (5.3) that p

(6.2) gijk = 'Okgij5 where (k' is a covariant vector. Substituting these expressions in (6.3) gijkl - gijlk = 0, which follows from (5.2) and (5.3),

we

obtain

a4f'l _'Pk Xk- axl*

VOL. 8, 1922

23

MA THEMA TICS: EISENHART AND VEBLEN

Hence the vector (Pk is the gradient of a scalar function, and can be put in the form log X (6.4)

Substituting this value of 'Pk in the right member of (6.2) and the explicit expression for gijk (cf. 2.4) in the left member, we obtain

(Xgi>) - Xgpi r,ik 1xgp r.

aak

-

=

(6.5)

o.

Consequently the functions Xgij satisfy (2.4) and give a Riemann geometry. It is interesting to note that the geometry of paths obtained from (6.2) without imposing the condition (6.4) is the geometry used by Weyl as the basis for a combined electromagnetic and gravitational theory. For (6.2) is equivalent to ak

rf

+gik (-i

=-(a -s12 \.x'

.xt

?.Xk / + 22 gik i+gJk (Pi -gis Yk). (6.6)

a

7. Let us now assume that (5.3) are algebraically consistent, and that all of their solutions satisfy (5.5). Let g(l), g. gt) be a complete set of solutions. The general solution is expressible in the form 71 (2) + i. + P(P). () + P(2) 911g (7.1) gjj = sP(l) gij gij= Differentiating (5.3) covariantly, we get (6.1), and consequently we must have (7.2) + )4(ap) g() (cf) = (al) gQ) + The p2 vectors X\(a (at, 13=1,....p) must be such that the functions gi7k satisfy (6.3). On substituting (7.2) in (6.3) we find

-*

bk

at +

k

(

(aT)- ) (a )Xk(f#) _

0.

(7.3)

-Y=1

When we express the condition that the functions gij given by (7.1) shall satisfy the conditions gijk = 0, we find in consequence of (7.2) that the functions (p( ) must satisfy the equations p

-(° Xk(a) = 0.

aSk ) +

(7.4)

~=1

In consequence of (7.3) this system of equations is completely integrable, and hence there exists a set of so's which by means of (7.1) determine a system of g's which yield a Riemann geometry. Hence we have the theorem: In order that the geometry of paths shall be a Riemann geometry it is sufficient that the r's be such that the equations (5.3) be algebraically consistent, and that all of their solutions satisfy (5.5).